Questions — OCR MEI C4 (332 questions)

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OCR MEI C4 2013 June Q1
1
  1. Express \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) }\) in partial fractions.
  2. Hence use binomial expansions to show that \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) } = a x + b x ^ { 2 } + \ldots\), where \(a\) and \(b\) are
    constants to be determined. State the set of values of \(x\) for which the expansion is valid.
OCR MEI C4 2013 June Q3
3 Using appropriate right-angled triangles, show that \(\tan 45 ^ { \circ } = 1\) and \(\tan 30 ^ { \circ } = \frac { 1 } { \sqrt { 3 } }\). Hence show that \(\tan 75 ^ { \circ } = 2 + \sqrt { 3 }\).
OCR MEI C4 2013 June Q4
4
  1. Find a vector equation of the line \(l\) joining the points \(( 0,1,3 )\) and \(( - 2,2,5 )\).
  2. Find the point of intersection of the line \(l\) with the plane \(x + 3 y + 2 z = 4\).
  3. Find the acute angle between the line \(l\) and the normal to the plane.
OCR MEI C4 2013 June Q5
5 The points \(\mathrm { A } , \mathrm { B }\) and C have coordinates \(\mathrm { A } ( 3,2 , - 1 ) , \mathrm { B } ( - 1,1,2 )\) and \(\mathrm { C } ( 10,5 , - 5 )\), relative to the origin O . Show that \(\overrightarrow { \mathrm { OC } }\) can be written in the form \(\lambda \overrightarrow { \mathrm { OA } } + \mu \overrightarrow { \mathrm { OB } }\), where \(\lambda\) and \(\mu\) are to be determined. What can you deduce about the points \(\mathrm { O } , \mathrm { A } , \mathrm { B }\) and C from the fact that \(\overrightarrow { \mathrm { OC } }\) can be expressed as a combination of \(\overrightarrow { \mathrm { OA } }\) and \(\overrightarrow { \mathrm { OB } }\) ?
OCR MEI C4 2013 June Q6
6 The motion of a particle is modelled by the differential equation $$v \frac { \mathrm {~d} v } { \mathrm {~d} x } + 4 x = 0 ,$$ where \(x\) is its displacement from a fixed point, and \(v\) is its velocity.
Initially \(x = 1\) and \(v = 4\).
  1. Solve the differential equation to show that \(v ^ { 2 } = 20 - 4 x ^ { 2 }\). Now consider motion for which \(x = \cos 2 t + 2 \sin 2 t\), where \(x\) is the displacement from a fixed point at time \(t\).
  2. Verify that, when \(t = 0 , x = 1\). Use the fact that \(v = \frac { \mathrm { d } x } { \mathrm {~d} t }\) to verify that when \(t = 0 , v = 4\).
  3. Express \(x\) in the form \(R \cos ( 2 t - \alpha )\), where \(R\) and \(\alpha\) are constants to be determined, and obtain the corresponding expression for \(v\). Hence or otherwise verify that, for this motion too, \(v ^ { 2 } = 20 - 4 x ^ { 2 }\).
  4. Use your answers to part (iii) to find the maximum value of \(x\), and the earliest time at which \(x\) reaches this maximum value.
OCR MEI C4 2013 June Q7
7 Fig. 7 shows the curve BC defined by the parametric equations $$x = 5 \ln u , y = u + \frac { 1 } { u } , \quad 1 \leqslant u \leqslant 10 .$$ The point A lies on the \(x\)-axis and AC is parallel to the \(y\)-axis. The tangent to the curve at C makes an angle \(\theta\) with AC, as shown. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4924020c-4df1-4bd5-aae9-95149a09f8c4-03_497_579_1612_744} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the lengths \(\mathrm { OA } , \mathrm { OB }\) and AC .
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(u\). Hence find the angle \(\theta\).
  3. Show that the cartesian equation of the curve is \(y = \mathrm { e } ^ { \frac { 1 } { 5 } x } + \mathrm { e } ^ { - \frac { 1 } { 5 } x }\). An object is formed by rotating the region OACB through \(360 ^ { \circ }\) about \(\mathrm { O } x\).
  4. Find the volume of the object. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR MEI C4 2014 June Q1
1 Express \(\frac { 3 x } { ( 2 - x ) \left( 4 + x ^ { 2 } \right) }\) in partial fractions.
OCR MEI C4 2014 June Q3
3 Fig. 3 shows the curve \(y = x ^ { 3 } + \sqrt { ( \sin x ) }\) for \(0 \leqslant x \leqslant \frac { \pi } { 4 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{413a0c52-b506-46d4-b1e4-fe13466abcc1-02_577_538_744_758} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Use the trapezium rule with 4 strips to estimate the area of the region bounded by the curve, the \(x\)-axis and the line \(x = \frac { \pi } { 4 }\), giving your answer to 3 decimal places.
  2. Suppose the number of strips in the trapezium rule is increased. Without doing further calculations, state, with a reason, whether the area estimate increases, decreases, or it is not possible to say.
OCR MEI C4 2014 June Q4
4
  1. Show that \(\cos ( \alpha + \beta ) = \frac { 1 - \tan \alpha \tan \beta } { \sec \alpha \sec \beta }\).
  2. Hence show that \(\cos 2 \alpha = \frac { 1 - \tan ^ { 2 } \alpha } { 1 + \tan ^ { 2 } \alpha }\).
  3. Hence or otherwise solve the equation \(\frac { 1 - \tan ^ { 2 } \theta } { 1 + \tan ^ { 2 } \theta } = \frac { 1 } { 2 }\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
OCR MEI C4 2014 June Q5
5 A curve has parametric equations \(x = \mathrm { e } ^ { 3 t } , y = t \mathrm { e } ^ { 2 t }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). Hence find the exact gradient of the curve at the point with parameter \(t = 1\).
  2. Find the cartesian equation of the curve in the form \(y = a x ^ { b } \ln x\), where \(a\) and \(b\) are constants to be determined.
OCR MEI C4 2014 June Q6
6 Fig. 6 shows the region enclosed by the curve \(y = \left( 1 + 2 x ^ { 2 } \right) ^ { \frac { 1 } { 3 } }\) and the line \(y = 2\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{413a0c52-b506-46d4-b1e4-fe13466abcc1-03_426_673_340_678} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} This region is rotated about the \(y\)-axis. Find the volume of revolution formed, giving your answer as a multiple of \(\pi\). \section*{Question 7 begins on page 4.}
OCR MEI C4 2014 June Q7
7 Fig. 7 shows a tetrahedron ABCD . The coordinates of the vertices, with respect to axes \(\mathrm { O } x y z\), are \(\mathrm { A } ( - 3,0,0 ) , \mathrm { B } ( 2,0 , - 2 ) , \mathrm { C } ( 0,4,0 )\) and \(\mathrm { D } ( 0,4,5 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{413a0c52-b506-46d4-b1e4-fe13466abcc1-04_794_844_456_589} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the lengths of the edges AB and AC , and the size of the angle CAB . Hence calculate the area of triangle ABC .
  2. (A) Verify that \(4 \mathbf { i } - 3 \mathbf { j } + 10 \mathbf { k }\) is normal to the plane ABC .
    (B) Hence find the equation of this plane.
  3. Write down a vector equation for the line through D perpendicular to the plane ABC . Hence find the point of intersection of this line with the plane ABC . The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × height.
  4. Find the volume of the tetrahedron ABCD .
OCR MEI C4 2014 June Q8
8 Fig. 8.1 shows an upright cylindrical barrel containing water. The water is leaking out of a hole in the side of the barrel. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{413a0c52-b506-46d4-b1e4-fe13466abcc1-05_254_442_347_794} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
\end{figure} The height of the water surface above the hole \(t\) seconds after opening the hole is \(h\) metres, where $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - A \sqrt { h }$$ and where \(A\) is a positive constant. Initially the water surface is 1 metre above the hole.
  1. Verify that the solution to this differential equation is $$h = \left( 1 - \frac { 1 } { 2 } A t \right) ^ { 2 } .$$ The water stops leaking when \(h = 0\). This occurs after 20 seconds.
  2. Find the value of \(A\), and the time when the height of the water surface above the hole is 0.5 m . Fig. 8.2 shows a similar situation with a different barrel; \(h\) is in metres. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{413a0c52-b506-46d4-b1e4-fe13466abcc1-05_232_447_1489_794} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
    \end{figure} For this barrel, $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - B \frac { \sqrt { h } } { ( 1 + h ) ^ { 2 } } ,$$ where \(B\) is a positive constant. When \(t = 0 , h = 1\).
  3. Solve this differential equation, and hence show that $$h ^ { \frac { 1 } { 2 } } \left( 30 + 20 h + 6 h ^ { 2 } \right) = 56 - 15 B t .$$
  4. Given that \(h = 0\) when \(t = 20\), find \(B\). Find also the time when the height of the water surface above the hole is 0.5 m . \section*{END OF QUESTION PAPER}
OCR MEI C4 2015 June Q2
2 Express \(6 \cos 2 \theta + \sin \theta\) in terms of \(\sin \theta\).
Hence solve the equation \(6 \cos 2 \theta + \sin \theta = 0\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
OCR MEI C4 2015 June Q3
3
  1. Find the first three terms of the binomial expansion of \(\frac { 1 } { \sqrt [ 3 ] { 1 - 2 x } }\). State the set of values of \(x\) for which
    the expansion is valid. the expansion is valid.
  2. Hence find \(a\) and \(b\) such that \(\frac { 1 - 3 x } { \sqrt [ 3 ] { 1 - 2 x } } = 1 + a x + b x ^ { 2 } + \ldots\).
OCR MEI C4 2015 June Q4
4 You are given that \(\mathrm { f } ( x ) = \cos x + \lambda \sin x\) where \(\lambda\) is a positive constant.
  1. Express \(\mathrm { f } ( x )\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving \(R\) and \(\alpha\) in terms of \(\lambda\).
  2. Given that the maximum value (as \(x\) varies) of \(\mathrm { f } ( x )\) is 2 , find \(R , \lambda\) and \(\alpha\), giving your answers in exact form.
OCR MEI C4 2015 June Q5
5 A curve has parametric equations \(x = \sec \theta , y = 2 \tan \theta\).
  1. Given that the derivative of \(\sec \theta\) is \(\sec \theta \tan \theta\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \operatorname { cosec } \theta\).
  2. Verify that the cartesian equation of the curve is \(y ^ { 2 } = 4 x ^ { 2 } - 4\). Fig. 5 shows the region enclosed by the curve and the line \(x = 2\). This region is rotated through \(180 ^ { \circ }\) about the \(x\)-axis. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{132ae754-bd4c-4819-80ef-4823ac2ead4f-02_545_853_1738_607} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure}
  3. Find the volume of revolution produced, giving your answer in exact form.
OCR MEI C4 2015 June Q6
6 Fig. 6 shows a lean-to greenhouse ABCDHEFG . With respect to coordinate axes Oxyz , the coordinates of the vertices are as shown. All distances are in metres. Ground level is the plane \(z = 0\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{132ae754-bd4c-4819-80ef-4823ac2ead4f-03_785_1283_424_392} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Verify that the equation of the plane through \(\mathrm { A } , \mathrm { B }\) and E is \(x + 6 y + 12 = 0\). Hence, given that F lies in this plane, show that \(a = - 2 \frac { 1 } { 3 }\).
  2. (A) Show that the vector \(\left( \begin{array} { r } 1
    - 6
    0 \end{array} \right)\) is normal to the plane DHC.
    (B) Hence find the cartesian equation of this plane.
    (C) Given that G lies in the plane DHC , find \(b\) and the length FG .
  3. Find the angle EFB . A straight wire joins point H to a point P which is half way between E and F . Q is a point two-thirds of the way along this wire, so that \(\mathrm { HQ } = 2 \mathrm { QP }\).
  4. Find the height of Q above the ground. \section*{Question 7 begins on page 4.}
OCR MEI C4 2015 June Q7
7 A drug is administered by an intravenous drip. The concentration, \(x\), of the drug in the blood is measured as a fraction of its maximum level. The drug concentration after \(t\) hours is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k \left( 1 + x - 2 x ^ { 2 } \right) ,$$ where \(0 \leqslant x < 1\), and \(k\) is a positive constant. Initially, \(x = 0\).
  1. Express \(\frac { 1 } { ( 1 + 2 x ) ( 1 - x ) }\) in partial fractions.
  2. Hence solve the differential equation to show that \(\frac { 1 + 2 x } { 1 - x } = \mathrm { e } ^ { 3 k t }\).
  3. After 1 hour the drug concentration reaches \(75 \%\) of its maximum value and so \(x = 0.75\). Find the value of \(k\), and the time taken for the drug concentration to reach \(90 \%\) of its maximum value.
  4. Rearrange the equation in part (ii) to show that \(x = \frac { 1 - \mathrm { e } ^ { - 3 k t } } { 1 + 2 \mathrm { e } ^ { - 3 k t } }\). Verify that in the long term the drug concentration approaches its maximum value. \section*{END OF QUESTION PAPER} \section*{Tuesday 16 J une 2015 - Afternoon} \section*{A2 GCE MATHEMATICS (MEI)} 4754/01B Applications of Advanced Mathematics (C4) Paper B: Comprehension \section*{QUESTION PAPER} \section*{Candidates answer on the Question Paper.} \section*{OCR supplied materials:}
    • Insert (inserted)
    • MEI Examination Formulae and Tables (MF2)
    \section*{Other materials required:}
    • Scientific or graphical calculator
    • Rough paper
    Duration: Up to 1 hour
    \includegraphics[max width=\textwidth, alt={}, center]{132ae754-bd4c-4819-80ef-4823ac2ead4f-05_117_495_1014_1308} PLEASE DO NOT WRITE IN THIS SPACE 2 In line 79 it says "For most journeys, more than half the journey time is composed of load time and transfer time". For what percentage of the journey time for the round trip made by car A in Table 4 is the car stationary?
    \includegraphics[max width=\textwidth, alt={}]{132ae754-bd4c-4819-80ef-4823ac2ead4f-07_645_1746_388_164}
    3 Using the expression on line 51, work out the answer to the question on lines 39 and 40 for the case where there are 10 upper floors and 7 people. Give your answer to 2 decimal places.
    \includegraphics[max width=\textwidth, alt={}]{132ae754-bd4c-4819-80ef-4823ac2ead4f-07_488_1746_1233_164}
    4 In lines 89 and 90 it says "... on average there will be approximately 8 stops per trip. A round trip with 8 stops would take between 188 and 200 seconds". Explain how the figure of 188 seconds has been derived. 5
  5. Referring to Strategy 3 and lines 99 to 101, complete the table below for car C .
  6. Calculate the time car C will take to transport all the people who work on floors 7 and 8 , and return to the ground floor.
    5
  		7. \includegraphics[max width=\textwidth, alt={}]{132ae754-bd4c-4819-80ef-4823ac2ead4f-08_1095_816_484_700}
    68 people make independent visits to any one of the upper floors of a building with 10 upper floors. What is the probability that at least one of the visitors goes to the top floor?
    6
    7 On lines 94 and 95 it says "Table 4 gives the timings for round trips in which the cars are required to stop at every floor they serve; Table 2 suggests this is a common occurrence in this case". Explain how Table 2 is used to make this claim.
    \includegraphics[max width=\textwidth, alt={}, center]{132ae754-bd4c-4819-80ef-4823ac2ead4f-09_1093_1740_1238_166} END OF QUESTION PAPER
OCR MEI C4 Q1
1 Explain why the number 1836.108 for the ratio Rest mass of electron would be suitable for communication with other civilisations whereas neither the rest mass of the proton nor that of the electron would be.
OCR MEI C4 Q2
2 A civilisation which works in base 5 sends out the first 6 digits of \(\pi\) as 3.032 32. Convert this to base 10.
OCR MEI C4 Q3
3 Complete this table to show the next 3 values of the iteration $$x _ { n + 1 } = k x _ { n } \left( 1 - x _ { n } \right)$$ in the case when \(k = 3.2\) and \(x _ { 0 } = 0.5\). Give your answers to calculator accuracy.
\(n\)\(x _ { n }\)
00.5
10.8
20.512
3
4
5
OCR MEI C4 Q4
4 Justify the statement that the equation in line 83, $$\frac { \phi } { 1 } = \frac { 1 } { \phi - 1 }$$ has the solution \(\phi = \frac { 1 \pm \sqrt { 5 } } { 2 }\).
OCR MEI C4 Q5
5 Justify the statement in line 87 that $$\frac { 1 } { \phi } = \frac { \sqrt { 5 } - 1 } { 2 }$$
OCR MEI C4 Q6
4 marks
6 A sequence is defined by $$a _ { n + 1 } = 2 a _ { n } + 3 a _ { n - 1 } \quad \text { with } a _ { 1 } = 1 \text { and } a _ { 2 } = 1 .$$ Using the method on page 5, show that the value to which the ratio of successive terms converges is 3 .
[0pt] [4]